1989 APMO Problems

Problem 1

Let $x_1, x_2, x_3, \dots , x_n$ be positive real numbers, and let

\[S=x_1+x_2+x_3+\cdots +x_n\].

Prove that

\[(1+x_1)(1+x_2)(1+x_3)\cdots (1+x_n)\leq 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots +\dfrac{S^n}{n!}\].


Problem 2

Prove that the equation


has no solutions in integers except $a=b=c=n=0$.


Problem 3

Let $A_1,A_2,A_3$ be three points in the plane, and for convenience, let $A_4=A_1$, $A_5=A_2$. For $n=1, 2,$ and $3$, suppose that $B_n$ is the midpoint of $A_nA_{n+1}$, and suppose that $C_n$ is the midpoint of $A_nB_n$. Suppose that $A_nC_{n+1}$ and $B_nA_{n+2}$ meet at $D_n$, and that $A_nB_{n+1}$ and $C_nA_{n+2}$ meet at $E_n$. Calculate the ratio of the area of triangle $D_1D_2D_3$ to the area of triangle $E_1E_2E_3$.


Problem 4

Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1\leq a<b\leq n$. Show that there are at least

\[4m\cdot \dfrac{\left(m-\dfrac{n^2}{4}\right)}{3n}\]

triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.


Problem 5

$f$ is a strictly increasing real-valued function on reals. It has inverse $f^{-1}$. Find all possible $f$ such that $f(x)+f'(x)=2x$ for all $x$.


See Also